muliple integral question

Question

Calculate the multiple integral : $\int\int\frac{1}{1+x^2}~dA$ where D is the closed triangular region with vertices (0,0), (1,1), and (0,1)?

So I have $\int_0^1\int_0^x \frac{1}{1+x^2}~dydx$ which gave me $\frac{ln(2)}{2}$ , but the answer is $\frac{\pi}{4}-\frac{\ln(2)}{2}$. Are my lower and upper bounds wrong? If i switch the y bounds to $\int_x^1$ then I get $\int_0^1\frac{1-x}{1+x^2}~dx$ which I have no idea how to integrate and whenIi put that into wolfram alpha I get $\frac{\pi}{4} - \frac{\ln(4)}{4}$ which is still wrong so where did I go wrong?

Answer 1

Redraw the region and switch the order: $A = \displaystyle \int_{0}^1 \int_{y}^1 \dfrac{1}{1+x^2} dxdy$. Then integrate with respect to $x$ first and integration by part with $y$ to finish it.